Number Of Qubits Research Articles

Classical variational optimization of a PREPARE circuit for quantum phase estimation of quantum chemistry Hamiltonians

We propose a method for constructing circuits for quantum phase estimation of a molecular Hamiltonian in quantum chemistry by using variational optimization of quantum circuits solely on classical computers. The circuit generates a quantum state which encodes the coefficients of the terms in the Hamiltonian as probability amplitudes and plays a crucial role in the state-of-the-art efficient implementations of quantum phase estimation. We employ the automatic quantum circuit encoding algorithm [Shirakawa , ] to construct circuits, which requires classical simulations of quantum circuits of O(logN) qubits with N being the number of qubits of the Hamiltonian. The generated circuits do not need any ancillary qubit. We demonstrate our method by investigating the number of T-gates of the obtained circuits for quantum chemistry Hamiltonians of various molecules, which shows a constant-factor reduction compared to previous approaches that do not use ancillary qubits. Since the number of available logical qubits and T gates will be limited at the early stage of the fault-tolerant quantum computing, the proposed method is particularly of use for performing the quantum phase estimation with such limited capability. Published by the American Physical Society 2024

Enhancing quantum field theory simulations on NISQ devices with Hamiltonian truncation

Quantum computers can efficiently simulate highly entangled quantum systems, offering a solution to challenges facing classical simulation of quantum field theories (QFTs). This paper presents an alternative to traditional methods for simulating the real-time evolution in QFTs by leveraging Hamiltonian truncation (HT). As a use case, we study the Schwinger model, systematically reducing the complexity of the Hamiltonian via HT while preserving essential physical properties. For the observables studied in this paper, the HT approach converges quickly with the number of qubits, allowing for the interesting physics processes to be captured without needing many qubits. Identifying the truncated free Hamiltonian’s eigenbasis with the quantum device’s computational basis avoids the need for complicated and costly state preparation routines, reducing the algorithm’s overall circuit depth and required coherence time. As a result, the HT approach to simulating QFTs on a quantum device is well suited to noisy-intermediate scale quantum devices, which have a limited number of qubits and short coherence times. We validate our approach by running simulations on a noisy-intermediate scale quantum device, showcasing strong agreement with theoretical predictions. We highlight the potential of HT for simulating QFTs on quantum hardware. Published by the American Physical Society 2024

Efficient Quantum Circuit Simulation by Tensor Network Methods on Modern GPUs

Efficient simulation of quantum circuits has become indispensable with the rapid development of quantum hardware. The primary simulation methods are based on state vectors and tensor networks. As the number of qubits and quantum gates grows larger in current quantum devices, traditional state-vector-based quantum circuit simulation methods prove inadequate due to the overwhelming size of the Hilbert space and extensive entanglement. Consequently, brutal force tensor network simulation algorithms become the only viable solution in such scenarios. The two main challenges faced in tensor network simulation algorithms are optimal contraction path finding and efficient execution on modern computing devices, with the latter determining the actual efficiency. In this study, we investigate the optimization of such tensor network simulations on modern GPUs and propose general optimization strategies from two aspects: computational efficiency and accuracy. First, we propose to transform critical Einstein summation operations into GEMM operations, leveraging the specific features of tensor network simulations to amplify the efficiency of GPUs. Second, by analyzing the data characteristics of quantum circuits, we employ extended precision to ensure the accuracy of simulation results and mixed precision to fully exploit the potential of GPUs, resulting in faster and more precise simulations. Our numerical experiments demonstrate that our approach can achieve a 3.96× reduction in verification time for random quantum circuit samples in the 18-cycle case of Sycamore, with sustained performance exceeding 21 TFLOPS on one A100. This method can be easily extended to the 20-cycle case, maintaining the same performance, accelerating by 12.5× compared to the state-of-the-art CPU-based results and 4.48–6.78× compared to the state-of-the-art GPU-based results reported in the literature. Furthermore, the strategies presented in this article hold general applicability for accelerating problems that can be tackled by contracting tensor networks such as graphical models and combinatorial optimizations.

Low-overhead quantum computing with the color code

Fault-tolerant quantum computation demands significant resources: large numbers of physical qubits must be checked for errors repeatedly to protect quantum data as logic gates are implemented in the presence of noise. We demonstrate that an approach based on the color code can lead to considerable reductions in the resource overheads compared with conventional methods, while remaining compatible with a two-dimensional layout. We propose a lattice surgery scheme that exploits the rich structure of the color-code phase to perform arbitrary pairs of commuting logical Pauli measurements in parallel while keeping the space cost low. Compared to lattice surgery schemes based on the surface code with the same code distance, and assuming the same amount of time is needed to complete a round of syndrome measurements, our approach yields about a 3× improvement in the space-time overhead, obtained from a combination of a 1.5× improvement in spatial overhead together with a 2× speedup due to the parallelization of commuting logical measurements. Even when taking into account the color code's lower error threshold using current decoders, the overhead is reduced by 10% at a physical error rate of 10−3 and by 50% at 10−4. Published by the American Physical Society 2024

Individually addressed entangling gates in a two-dimensional ion crystal

Two-dimensional (2D) ion crystals may represent a promising path to scale up qubit numbers for ion trap quantum information processing. However, to realize universal quantum computing in this system, individually addressed high-fidelity two-qubit entangling gates still remain challenging due to the inevitable micromotion of ions in a 2D crystal as well as the technical difficulty in 2D addressing. Here we demonstrate two-qubit entangling gates between any ion pairs in a 2D crystal of four ions. We use symmetrically placed crossed acousto-optic deflectors (AODs) to drive Raman transitions and achieve an addressing crosstalk error below 0.1%. We design and demonstrate a gate sequence by alternatingly addressing two target ions, making it compatible with any single-ion addressing techniques without crosstalk from multiple addressing beams. We further examine the gate performance versus the micromotion amplitude of the ions and show that its effect can be compensated by a recalibration of the laser intensity without degrading the gate fidelity. Our work paves the way for ion trap quantum computing with hundreds to thousands of qubits on a 2D ion crystal.

Molecular electronic structure calculation via a quantum computer

Quantum computers can be used to calculate the electronic structure and estimate the ground state energy of many-electron molecular systems. In the present study, we implement the Variational Quantum Eigensolver (VQE) algorithm, as a hybrid quantum–classical algorithm to calculate the ground state energy of the molecules such as H3+, OH−, HF and BH3 in which the number of qubits has an increasing trend. We use the parity transformation for fermion to qubit encoding and the Unitary Coupled Cluster for Single and Double excitations (UCCSD) to construct an ansatz. We compare our quantum simulation results with the computational chemistry approaches including Full Configuration Interaction (FCI), as benchmark energy and Unrestricted Hartree–Fock (UHF), as a common computational method. Our results show that there is a good agreement between molecular ground state energy obtained from VQE and FCI. Moreover, the accuracy of the ground state energies obtained from VQE in our work is higher than the previously reported values. This work aims to benchmark the VQE algorithm to calculate the electronic ground state energy for a new set of molecules that can be good candidates for molecular simulation on a real quantum computer.

Layer codes

Quantum computers require memories that are capable of storing quantum information reliably for long periods of time. The surface code is a two-dimensional quantum memory with code parameters that scale optimally with the number of physical qubits, under the constraint of two-dimensional locality. In three spatial dimensions an analogous simple yet optimal code was not previously known. Here we present a family of three dimensional topological codes with optimal scaling code parameters and a polynomial energy barrier. Our codes are based on a construction that takes in a stabilizer code and outputs a three-dimensional topological code with related code parameters. The output codes are topological defect networks formed by layers of surface code joined along one-dimensional junctions, with a maximum stabilizer check weight of six. When the input is a family of good quantum low-density parity-check codes the output codes have optimal scaling. Our results uncover strongly-correlated states of quantum matter that are capable of storing quantum information with the strongest possible protection from errors that is achievable in three dimensions.

A quantum computing measurement and control system with an FPGA-based scheduling system.

Large-scale qubit systems represent the future of superconducting quantum computing. As the number of qubits and functions increases, the current qubit control architectures struggle to achieve large-scale expansion. To address this issue, we have developed a field-programmable gate array-based scheduling system, which serves as a control architecture deployed on measurement and control hardware. This system achieves the generation of multi-channel long sequence pulses using minimal storage space, with reusable envelope and instruction designs. The envelopes and instructions are preloaded into the electronics, with each instruction corresponding to a different segment of the output waveform. The architecture we designed allows for extended pulse control lengths and more complex experiments without the requirement for external storage resources. We present the resource utilization of the implementation and compare it with previous measurements under different qubit counts. To verify the functionality of the system, we conducted a series of performance tests and characterization experiments on the deployed electronics. The measured single-qubit relaxation time T1 is 34.1 μs, and the dephasing time T2* is 25.8 μs. We characterized the quality of the control pulses using a randomized benchmarking protocol, achieving an average fidelity of 99.962%.

Opening the Black Box inside Grover’s Algorithm

Grover’s algorithm is one of the primary algorithms offered as evidence that quantum computers can provide an advantage over classical computers. It involves an “oracle” (external quantum subroutine), which must be specified for a given application and whose internal structure is not part of the formal scaling of the quadratic quantum speedup guaranteed by the algorithm. Grover’’s algorithm also requires exponentially many calls to the quantum oracle (approximately 2n calls where n is the number of qubits) to succeed, raising the question of its implementation on both noisy and error-corrected quantum computers. In this work, we construct a quantum-inspired algorithm executable on a classical computer that performs Grover’s task in a linear number of calls to (simulations of) the oracle—an exponentially smaller number than Grover’s algorithm—and demonstrate this algorithm explicitly for Boolean satisfiability problems. The complexity of our algorithm depends on the cost to simulate the oracle once, which may or may not be exponential, depending on its internal structure. Indeed, Grover’s algorithm does not have an quantum speedup as soon as one is given access to the “source code” of the oracle, which may reveal an internal structure of the problem. Our findings illustrate this point explicitly, as our algorithm exploits the structure of the quantum circuit used to program the quantum computer to speed up the search. There are still problems where Grover’s algorithm would provide an asymptotic speedup if it could be run accurately for large enough sizes. Our quantum-inspired algorithm provides lower bounds, in terms of the quantum-circuit complexity, for the quantum hardware to beat classical approaches for these problems. These estimates, combined with the unfavorable scaling of the success probability of Grover’s algorithm, which in the presence of noise decays as the exponential of the exponential of the number of qubits, makes a practical speedup unrealistic even under extremely optimistic assumptions of the evolution of both hardware quality and availability. Published by the American Physical Society 2024

Scalable Multispecies Ion Transport in a Grid-Based Surface-Electrode Trap

Quantum processors based on linear arrays of trapped ions have achieved exceptional performance, but scaling to large qubit numbers requires realizing two-dimensional ion arrays as envisioned in the quantum charge-coupled device (QCCD) architecture. Here, we present a scalable method for the control of ion crystals in a grid-based surface-electrode Paul trap and characterize it in the context of transport operations that sort and reorder multispecies crystals. By combining cowiring of control electrodes at translationally symmetric locations in each grid site with the sitewise ability to exchange the voltages applied to two special electrodes gated by a binary input, site-dependent operations can be achieved using only a fixed number of analog voltage signals and a single digital input per site. In two separate experimental systems containing nominally identical grid traps, one using Yb+171−Ba+138 crystals and the other Ba+137−Sr+88, we demonstrate this method by characterizing the conditional intrasite crystal reorder and the conditional exchange of ions between adjacent sites on the grid. Averaged across a multisite region of interest, we measure subquanta motional excitation in the axial in-phase and out-of-phase modes of the crystals following these operations at exchange rates of 2.5 kHz. In this initial demonstration, the logic controlling the voltage exchange occurs in software, but the applied signals mimic a proposed hardware implementation using crossover switches. These techniques can be further extended to implement other conditional operations in the QCCD architecture such as gates, initialization, and measurement. Published by the American Physical Society 2024

Solving transport equations on quantum computers—potential and limitations of physics-informed quantum circuits

AbstractQuantum circuits with trainable parameters, paired with classical optimization routines can be used as machine learning models. The recently popularized physics-informed neural network (PINN) approach is a machine learning algorithm that solves differential equations by incorporating them into a loss function. Being a mesh-free method, it is a promising approach for computational fluid dynamics. The question arises whether the properties of quantum circuits can be leveraged for a quantum physics-informed machine learning model. In this study, we compare the classical PINN-ansatz and its quantum analog, which we name the physics-informed quantum circuit (PIQC). The PIQC simulations are performed on a noise-free quantum computing simulator. Studying various differential equations, we compare expressivity, accuracy and convergence properties. We find that one-dimensional problems, such as the linear transport of a Gaussian-pulse or Burgers’ equation, allow a successful approximation with the classical and the quantum ansatz. For these examples, the PIQC overall performs similarly to PINN and converges more consistently and for Burgers’ equations even faster. While this is promising, the chosen quantum circuit approach struggles to approximate discontinuous solutions which the classical PINN-ansatz can represent. Based on this comparison, we extrapolate that the required number of qubits for solving two-dimensional problems in aerodynamics may already be available in the next few years. However, the acceleration potential is currently unclear and challenges like noisy circuits and approximations of discontinuous solutions have to be overcome.

Improvements to Quantum Interior Point Method for Linear Optimization

Quantum linear system algorithms (QLSA) have the potential to speed up Interior Point Methods (IPM). However, a major bottleneck is the inexactness of quantum Tomography to extract classical solutions from quantum states. In addition, QLSAs are sensitive to the condition number, and this sensitivity is exacerbated when the Newton systems arising in IPMs converge to a singular matrix. Recently, an Inexact Feasible Quantum IPM (IF-QIPM) has been developed that addresses the inexactness of QLSAs. However, this method requires a large number of gates and qubits to be implemented. Here, we propose a new IF-QIPM using the normal equation system, which requires less number of gates and qubits. To mitigate the sensitivity to the condition number and other input data-related parameters, we use preconditioning coupled with iterative refinement to obtain better complexity. Finally, we demonstrate the effectiveness of our approach on IBM Qiskit simulators.

A qubit-efficient variational selected configuration-interaction method

Abstract Finding the ground-state energy of molecules is an important and challenging computational problem for which quantum computing can potentially find efficient solutions. The variational quantum eigensolver (VQE) is a quantum algorithm that tackles the molecular groundstate problem and is regarded as one of the flagships of quantum computing. Yet, to date, only very small molecules were computed via VQE, due to high noise levels in current quantum devices. Here we present an alternative variational quantum scheme that requires significantly less qubits than VQE. The reduction in the qubit number allows for shallower circuits to be sufficient, rendering the method more resistant to noise. The proposed algorithm, termed variational quantum selected-configuration-interaction (VQ-SCI), is based on: (a) representing the target groundstate as a superposition of Slater determinant configurations, encoded directly upon the quantum computational basis states; and (b) selecting a-priory only the most dominant configurations. This is demonstrated through a set of groundstate calculations of the H2, LiH, BeH2, H2O, NH3 and C2H4 molecules in the sto-3g basis set, performed on IBM quantum devices. We show that the VQ-SCI reaches the full configuration interaction energy within chemical accuracy using the lowest number of qubits reported to date. Moreover, when the SCI matrix is generated ‘on the fly’, the VQ-SCI requires exponentially less memory than classical SCI methods. This offers a potential remedy to a severe memory bottleneck problem in classical SCI calculations. Finally, the proposed scheme is general and can be straightforwardly applied for finding the groundstate of any Hermitian matrix, outside the chemical context.

Applying quantum approximate optimization to the heterogeneous vehicle routing problem

Quantum computing offers new heuristics for combinatorial problems. With small- and intermediate-scale quantum devices becoming available, it is possible to implement and test these heuristics on small-size problems. A candidate for such combinatorial problems is the heterogeneous vehicle routing problem (HVRP): the problem of finding the optimal set of routes, given a heterogeneous fleet of vehicles with varying loading capacities, to deliver goods to a given set of customers. In this work, we investigate the potential use of a quantum computer to find approximate solutions to the HVRP using the quantum approximate optimization algorithm (QAOA). For this purpose we formulate a mapping of the HVRP to an Ising Hamiltonian and simulate the algorithm on problem instances of up to 21 qubits. We show that the number of qubits needed for this mapping scales quadratically with the number of customers. We compare the performance of different classical optimizers in the QAOA for varying problem size of the HVRP, finding a trade-off between optimizer performance and runtime.

Chemical Reaction Simulator on Quantum Computers by First Quantization (II)─Basic Treatment: Implementation.

Chemical simulation is a key application area that can leverage the power of quantum computers. A chemical simulator that implements a grid-based first quantization method has promising characteristics, but an implementation fully in quantum circuits seems to have not been published. Here, we present "crsQ" (chemical reaction simulator Q), which is a quantum circuit generator that generates such a chemical simulator. The generated simulator is capable of antisymmetrization of the initial wave function and time-evolution of the wave function based on the Suzuki-Trotter decomposition. The potential energy term of the Hamiltonian is implemented using arithmetic gates, such as adders, subtractors, multipliers, dividers, and square roots. Circuit diagrams and output samples are shown. The number of qubits in the circuits scales on the order of O(η log η), where η is the number of electrons. Each component of the generated circuit was verified in unit tests. Along with this development, we designed frameworks to ease the development of large-scale circuits, namely, a temporary qubit allocation framework and an abstract syntax tree framework for arithmetic formulas. These frameworks are expected to be useful in large-scale quantum circuit generators.

Shallow-Depth Quantum Circuit for Unstructured Database Search

Grover’s search algorithm (GSA) offers quadratic speedup in searching unstructured databases but suffers from exponential circuit depth complexity. Here, we present two quantum circuits called HX and Ry layers for the searching problem. Remarkably, both circuits maintain a fixed circuit depth of two and one, respectively, irrespective of the number of qubits used. When the target element’s position index is known, we prove that either circuit, combined with a single multi-controlled X gate, effectively amplifies the target element’s probability to over 0.99 for any qubit number greater than seven. To search unknown databases, we use the depth-1 Ry layer as the ansatz in the Variational Quantum Search (VQS), whose efficacy is validated through numerical experiments on databases with up to 26 qubits. The VQS with the Ry layer exhibits an exponential advantage, in circuit depth, over the GSA for databases of up to 26 qubits.

Option pricing under stochastic volatility on a quantum computer

We develop quantum algorithms for pricing Asian and barrier options under the Heston model, a popular stochastic volatility model, and estimate their costs, in terms of T-count, T-depth and number of logical qubits, on instances under typical market conditions. These algorithms are based on combining well-established numerical methods for stochastic differential equations and quantum amplitude estimation technique. In particular, we empirically show that, despite its simplicity, weak Euler method achieves the same level of accuracy as the better-known strong Euler method in this task. Furthermore, by eliminating the expensive procedure of preparing Gaussian states, the quantum algorithm based on weak Euler scheme achieves drastically better efficiency than the one based on strong Euler scheme. Our resource analysis suggests that option pricing under stochastic volatility is a promising application of quantum computers, and that our algorithms render the hardware requirement for reaching practical quantum advantage in financial applications less stringent than prior art.

On the stabilizer formalism and its generalization

Abstract The standard stabilizer formalism provides a setting to show that quantum computations restricted to operations within the Clifford group are classically efficiently simulable: this is the content of the well-known Gottesman–Knill theorem. This work analyzes the mathematical structure behind this theorem to find possible generalizations and derivation of constraints required for constructing a non-trivial generalized Clifford group. We prove that if the closure of the stabilizing set is dense in the set of SU(d) transformations, then the associated Clifford group is trivial, consisting only of local gates and permutations of subsystems. This result demonstrates the close relationship between the density of the stabilizing set and the simplicity of the corresponding Clifford group. We apply the analysis to investigate stabilization with binary observables for qubits and find that the formalism is equivalent to the standard stabilization for a low number of qubits. Based on the observed patterns, we conjecture that a large class of generalized stabilizer states are equivalent to the standard ones. Our results provide better insights into the structure of Gottesman–Knill-type results, consequently allowing us to draw a sharper line between quantum and classical computation.

Scalable Architecture for Trapped-Ion Quantum Computing Using rf Traps and Dynamic Optical Potentials

Qubits based on ions trapped in linear radio-frequency traps form a successful platform for quantum computing, due to their high fidelity of operations, all-to-all connectivity, and degree of local control. In principle, there is no fundamental limit to the number of ion-based qubits that can be confined in a single 1D register. However, in practice, there are two main issues associated with long trapped-ion crystals, that stem from the “softening” of their modes of motion, upon scaling up: high heating rates of the ions’ motion and a dense motional spectrum; both impede the performance of high-fidelity qubit operations. Here, we propose a holistic, scalable architecture for quantum computing with large ion crystals that overcomes these issues. Our method relies on dynamically operated optical potentials that instantaneously segment the ion crystal into cells of a manageable size. We show that these cells behave as nearly independent quantum registers, allowing for parallel entangling gates on all cells. The ability to reconfigure the optical potentials guarantees connectivity across the full ion crystal and also enables efficient midcircuit measurements. We study the implementation of large-scale parallel multiqubit entangling gates that operate simultaneously on all cells and present a protocol to compensate for crosstalk errors, enabling full-scale usage of an extensively large register. We illustrate that this architecture is advantageous both for fault-tolerant digital quantum computation and for analog quantum simulations. Published by the American Physical Society 2024

Preparation of hybrid W entangled states between superconducting qubits and microwave resonators in circuit QED

Hybrid W entangled states are essential in quantum information processing, quantum communication, and quantum technology. In this Letter, we propose a simple method to prepare hybrid W entangled states between n superconducting (SC) qubits and n microwave resonators (MRs) in circuit QED. Only two basic operations are needed for the preparation of hybrid W states. The operational time decreases as the number of qubits increases. Since no ancillary cavity is required, the hardware resources for the state preparation are minimized. Because the state preparation does not involve any measurements, the hybrid W entangled states are generated in a deterministic way. Moreover, during the entire preparation, the high-energy levels of the SC qutrits remain unexcited, which greatly reduces decoherence of the SC qutrits. As an example, our numerical simulation demonstrates that high-fidelity preparation of the hybrid W entangled state of three SC qubits and three MRs is feasible within the current circuit QED technique. This method is universal and can be applied to generate hybrid W states of n matter qubits (e.g., atomic qubits, NV center qubits, quantum dot qubits, and magnon qubits) and n photonic qubits in various physical systems.